Purification of a density operator

A purification of a density operator ${\rho }_A$ is a pure state $|\Psi {⟩}_{AE}$ on a larger system such that ${\mathrm{Tr}}_E|\Psi ⟩⟨\Psi |={\rho }_A$. A spectral decomposition ${\rho }_A={\sum }_j{p}_j|j⟩⟨j|$ gives the canonical construction $|\Psi ⟩={\sum }_j\sqrt{p_j}|j{⟩}_A|j{⟩}_E$.

Any two purifications of the same density operator on sufficiently large reference spaces differ by an isometry on the reference system, and by a unitary when the reference spaces have the same dimension.

The Hughston--Jozsa--Wootters theorem says that two pure-state ensembles represent the same density operator exactly when their subnormalized state vectors are related by an isometry, or by a unitary after padding the shorter ensemble with zero vectors.

Uhlmann's theorem states that the fidelity of two density operators is the maximum absolute overlap between their purifications on a common reference space.